RELATIVE MODEL-COMPLETENESS AND THE ELIMINATION OF QUANTIFIERS

RELATIVE MODEL-COMPLETENESS AND THE ELIMINATION OF QUANTIFIERS

by Abraham ROBINSON, Jerusalem

Dedication

The results of the present paper hinge on the extended first &heorem of Hilbert-Bernays (ref. 1). I t seems fitting to recall in this manner tha t many of the highly specialized results OF contem- porary Mathematical Logic rely to a large extent on the work of Paul Rernays as incorporated in the two volumes of the classic just quoted, as well as on his earlier contributions t o the foundations of the lower predicate calculus.

1. Introducfion

I t is known that for a model-complete system of axioms in the lower predicate calculus, every predicate is equivalent t o an existential predicate (i.e. a predicate in prenex normal form with existential quantifiers only, ref. 4). Conversely (ref. 6) , if with respect to a given system of asioms K, every predicate which is defined in K is equivalent to an existential predicate then K is model-complete. Moreover, if I< is model-complete then, by defi- nition, the set K u N is complete, where N is the diagram of any given model M of K (i.e. the set of atomic formulce, and of the negations of such formulz, which hold in M). It then follows that if K and ;tz are given in a suficiently concrete manner, K recursive and M computable, say (making N recursive), then K u N is decidable. More particularly, if K possesses a prime model then K alone is complete and decidable.

1.A summary of a talk on the subject of this paper will be found in Summaries of talks presented ut the Summer Institute of Symbolic Logic at Cornell University, 1957, vol. I , pp. 155-158.

RELATIVE MODEL-COMPLETENESS 395

These facts are proved in the above-mentioned references for theories with relations and individual constants only but it is not difficult to see that they are equally true when functors are included. The proofs are in fact carried out without any detailed reference to the particular form or content of the axioms. We observe only that a diagram, still defined as above, may now contain sentences of the form - R (p (a, b) , y (c)), where 9, y are functors and R is a relation.

By way of contrast, the so-called method of elimination of quantifiers depends intimately on the mathematical properties of the particular system under consideration. When available in an effective form, this method provides a decision procedure and, in some cases, a completeness proof. However, its realization may still require considerable ingenuity (e. g. ref. 8) so that in general it represents a suggestion, or piece of good advice, rather than a systematic method. In its simplest form, the suggestion is that to any given existential predicate

Q (XI, - * * , xn) = ( 3 ~ 1 ) * . . ( g y m ) Z (XI, * - 9 x n , ~ 1 , * * - 9 ~ m )

where Z is free of quantifiers, we try to find another predicate, Q' (xl, . . . , xn), which is free of quantifiers such that the equivalence

(xi) . - - (xn) [Q (21, . - .) 2") = Q' (xi, - . * , xn)]

is deducible from K, briefly

K t- (XI) . * * (xn ) [Q (XI, - - * , xn) EE Q' (XI, * * , xn)].

If such predicates Q' can be found for all existential predicates Q, then it is an easy matter to do as much for all universal predicates (predicates in prenex normal form with universal quantifiers only) and hence, for all predicates and sentences.

To show that the elimination of quantifiers from existential predicates is in fact sufficient, let us consider a simple example. Let

X = (Y) (32) (w) Q (Y, ~3 W)

where Q does not contain any quantifiers. (m) Q (y, z, m), of y and z.

Consider the predicate Its negation is, reduced to normal

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form, (3w) - Q (y, z , w). without quantifiers, such tha t

Hence

Again, by assumption,

By assumption, there exists a Q, (y, z) ,

K t (y) (2) [[(a W) - Q (Y, Z, ~ ) l K I- X = (Y) (3%) - Q, (Y, 2).

Qi (Y, 211 *

K t- (Y) “(32) - Qi (Y, z)l Q 2 (Y)I for some Q2 (y) without quantifiers, and so

K t X (Y) Qz (Y). Finally, for some Q3 without quantifiers

K t ( 3 ~ ) - Qz (Y) and so

K t X r - Q ,

Q3 is the required sentence without quantifiers. In order to iink up the theory of model-completeness with the

method of elimination of quantifiers we ask ourselves under what conditions a given predicate can be reduced not only t o an existential predicate, but even to a predicate which is free of quantifiers altogether. Or, in view of what has just been said, under what conditions can an existential predicate be replaced by one free of quantifiers. I t is clear that , in general, this will be possible only if the calculus is endowed with functors.

All sentences will be supposed closed.

Q3

We regard individual constants as functors of order zero.

2.

Let

Elimination of quanfifiers from invariant predicates

Q (q, . . ., xn), n 2 0,

be a predicate which is defined in a consistent set of axioms K (i. e. all the extralogical constants of Q appear in K). Suppose that K consists of sentences in prenes normal form with universal quantifiers only but contains a t least one individual constant if n = O .


Suppose that Q (x,, . . ., xn) is invariant with respect to K. (A predicate Q is invariant with respect to K if both Q and - Q are persistent with respect to K. Q is persistent with respect to K if whenever Q (a,, . . ., a,) holds in a model M of K, a,, . . . , an E M, then Q (a,, . . ., a,) holds also in all extensions of M which are models of K).

2.1. Theorem. Under the above assumptions, there exists a predicate Q’ (x,, . . ., 2,) which is defined in K and which is free of quantifiers such that

K t (q) . * * ( x n ) [Q (21, * * , xn) E Q’ (XI, - * , x n ) ] .

To prove this theorem we make use of the extended first &-theorem of Hilbert-Bernays (ref. 1, p. 66) whose usefulness for the applications of logic to algebra was first pointed out by G. Kreisel (ref. 2). For our present purpose, this theorem may be stated as follows.

2.2. Suppose an existential statement

-Z free of quantifiers-is deducible from a set of universal state- ments X , , . . ., X k which contain a t least one individual constant. Then there exist terms ti of X , , . . ., X k (Le. expressions obtained from the functors of X , , . . ., Xk by repeated application, e.g. rp (a, y (a, b) , c)) such that the sentence

Y = z ( l i , . . ., th) v z (212, . . ., t&) v. . . v z ( I ? , . . ., tk) is deducible from x,, . . ., x k .

We note that Y 1 X is provable. The inclusion of a t least one individual constant in X,, . . ., X k obviates the necessity of going beyond the extralogical constants of X , , . . . , Xk in the formulation of the terms ti. This can be seen without difficulty by means of Herbrand’s construction.

Proof of 2.1. Since the predicate Q (x,, . . ., xn) which occurs in the theorem is invariant there exist (compare refs. 3, 6, 9) existential predicates

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Note tha t 2.4 is equivalent to

2.5. K (xJ . . . (5,) [Q (z1, . . ., z,) zz [(Z1) . . . ( z ~ ) - - z, (q, * * * 9 %I, 21, * . - 1 z , ) ]] .

Introducing individual constants aI, . . ., (I, which did not occur previously we then have

2.6. K i- [(q) . . . (z l ) - Z, (al, . . ., a,, zl, . . ., zl ) ] 3

>[(3y1) . * . ( 2 y m ) z (a19 * * . Y a n , .%, * * * grn)]-

Using the extended completeness theorem of the lower predicate calculus we conclude from 2.6 t ha t for a certain finite subset {XI , . . ., X,) of K, k 2 0, t he sentence

( 3 ~ ~ ) - * (39,) Z1 (a19 . . ., an, 91, 9 . 4 3 9,)

is deducible from

2.7. XI, . . . , Xk J (zl) . . . (q) - z, (a,, . . . , a,, z,, . . . , Z l ) 1. I t now follows from the extended first &-theorem tha t there are terms 14 involving some or all of the functors of X,, . . ., X k as well as of Z,(a,, . . ., a,,, zl, . . ., z I ) , such t h a t

z, ( 0 1 , . . ., a l l , 6, . . ., I:,$) \/. . . v z, (a1, . . ., a,, $, . I ., Ik,)

is deducible from 2.i. Hence


where

Q’ (q, . . . , x,,) = Z, (x~, . . . , zn, t: . . . th) V . . . V Z , (x,, . . . , x n , T? . . . t:) and where the t i are obtained from the t!; by replacing a,, . . ., a,, everywhere by q, . . ., 2,. On the other hand,

is provable, and so

where Q’ (x,, . . . , x,,) has the form required by 2.1. This estab- lishes the t ruth of the theorem.

Since all predicates defined in a model-complete set of axioms K are persistent with respect to K, and hence, are invariant with respect to K, and since every set of axioms can be replaced by a set of axioms in prenex normal form with universal quantifiers only, by the introduction of Herbrand functors (Skolem functors, Hilbert choice-functions), i t would appear tha t we have already achieved our aim of proving the eliminability of quantifiers for an important class of systems. However, when we try t o apply our result t o a concrete case we see that it is far from satisfactory. Consider in particular the theory of real-closed ordered fields for which a method of elimination was worked out by Tarski in a well- known paper (ref. 8). Let KO be a set of axioms for the concept of a real-closed ordered field, formulated, to begin with, in terms of the relations of addition, multiplication, equality, and order, (S (2, y, z ) , P (x, y, z), E (x, y), Q (5, y)) and without functors. In order to replace KO by a set of axioms to which theorem 2.1 can he applied, we have to introduce, first of all, functors for addition and multiplication and individual constants for zero and one. This is only natural and might as well have been done from the outset. However, in addition, and in view of the axioms which affirm the existence of difference and quotient, of the square roots of positive numbers, and of roots of equations of odd degree we also have to introduce corresponding functors. The resulting set of sentences is, in the first instance, incomplete, owing to the introduction of

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new extralogical constants but this state of affairs can be remedied by the introduction of additional axioms which make the functors just mentioned unique, e.g. by selecting positive square roots, and smallest roots of polynomials of odd degree for the corresponding functors. A more critical blemish of our method is tha t these new functors will, in general, enter into the predicates Q’ (q, . . . , 2,) of theorem 2.1, which may accordingly involve square roots and roots of equations of high order with xl, . . . , x,, as parameters. This compares unfavourably with the Q’ (zl, . . . , z,) provided by Tarski’s procedure, which involves only the rational operations. However, i t turns out that we can save the situation by making use of the notion of relative model-completeness which was first intro- duced elsewhere (ref. 7).

3. Relative model-completeness

The following definitions and results have been given (ref. 7) for theories without functors of positive order (i.e. other than individual constants), but they can be formulated and proved equally well when such functors are included.

We consider two non-empty and consistent sets of sentences K and K* concerning which the following assumptions will be made throughout this section.

K* does not include any extra-logical constants which are not also included in K.

K* is model-consistent with K, tha t is t o say, every model of K can be embedded in a model of K*, K* u N is consistent for the diagram N of any model M of K.

3.1.

3.2.

3.3. This condition is somewhat stronger than the corresponding

condition 2.3 in ref. 7 , but includes all cases which are of interest in connection with our present discussion.

We say that K* is model-complete relative to K if for any model &I of K, the set K* u N is complete, where N is the diagram of M. Or, t o put i t in a different way, any two extensions of a model M of K which are models of K* satisfy the same elementary

K is deducible from K*, e.g. K C K*.


theorems which are formulated in terms of the relations and functors of M.

There is a test for relative model-completeness which is quite similar to the test for ordinary model-completeness (ref. 4). However, for practical purposes it may be easier to establish the absolute model-completeness of K* and then to prove model- completeness relative to the given K by means of the following result.

3.4. If K* is model-complete, and if for any model M of K with diagram N, K* u N possesses a prime model M, (i.e. a structure which is such that any other model of K* u N possesses a partial structure which is isomorphic to M, by an isomorphism which centralizes the elements of M) then K* is model-complete relative to K.

The following examples will clarify these notions. They will also be useful in the sequel.

(i) Let K be a set of axioms for the concept of an integral domain with distinct zero and unit elements and formulated in terms of the relation of equality E (2, y) and in term of the operation of addition, (T (5, y) (i.e. z + y), multiplication 7c ($, y) (i.e. q), the inverse operation, p (2) (i.e. - x) and in terms of the individual constants 0 and 1. It is not difficult to check that when all these extralogical constants have been included, the axioms of K, including the axioms of equivalence and substitutivity for E ( , ), can be formulated in prenex normal form with universal quantifiers only, e.g.

We now enlarge K to yield a set of axioms K* for the concept of an algebraically closed field, without however introducing any new extralogical constants. This requires the inclusion, first, of a law of divisibility, except by 0, and next, of a sequence of axioms postulating the solubility of equations of higher order. When written in prenex normal form these axioms will include both universal and existential quantifiers.

We note that conditions 3.1-3.3 are all satified. 13

402 A. ROBINSON

I t is known that the concept of an algebraically closed field is model-complete. Although the set of axioms for which this fact is established in ref. 4 is based on a different set of extralogical constants, it is easily seen that the arguments and conclusion apply equally well to the present K*.

Now let M be any model of K, i.e. any integral domain with distinct 0 and 1. In order to obtain a model M, of K* u M as required in the theorem 3.5 we only have to extend M first to its field of fractions (quotients) and then to the algebraic closure of that field. We conclude that K* is model-complete relative to K.

(ii) Let K be a set of axioms for the concept of a commutative ordered ring in which 0 and 1 are distinct. K may be formulated in terms of the extralogical constants of ( i ) together with the relation Q (x, y) (i.e. 2 5 y) and may again be assumed to consist of axioms in prenex normal form with universal quantifiers only. Let K* be a set of axioms for the concept of a real-closed ordered field obtained by enlarging K without adding any new extralogical constants.

M is an integral domain since it is an ordered ring and can be extended to its ordered field of fractions and further to the real cIosure of that field, M,,. Then 3.1-3.3 hold and moreover, M, satisfies the conditions of 3.5. I t follows that K* is model-complete relative to K.

To conclude this section, we note that if a set K* is model- complete relative to a set K, and if 3.3 is satisfied, then K* is model-complete in the absolute (i.e. ordinary) sense.

Then K* is model-complete (ref. 4). Let M be a model of K.

4. Relative model-completeness and the elimination of quantifiers

Let K* and K be two non-empty and consistent sets of state- ments. A predicate Q* (x,, . . ., x,), n 2 0, which is defined in K is said to be invariant with respect fo K* over K, if for any set of elements a,, . . ., a, of a model M of K, either Q* (a,, . . ., a,) holds in all extensions of M which are models of K* or - Q* (a,, . . ., a,) holds in all extension of hl which are models of K*.

4.1. Theorem. Let K, K* be non-empty and consistent such that 3.1 and 3.2 are satisfied, and let Q* (q, . . . , 5,) be a predicate


which is defined in K* and is invariant with respect to K* over K. Then there exists a predicate Q (xl, . . ., 2,) which is defined in K such that for any set a,, . . ., a, of individual constants in a model M of K, Q (al, . . ., a,) holds in M if and only if Q* (al, . . ., a,) holds in all models of K* which are extensions of M. Q is said to be a (or the) projection of Q* from K* t o K.

4.2. Theorem. Let K, K* be non-empty and consistent, and satisfying 3.1, 3.2, such that K* is model-complete relative to K. Then to every predicate Q* (q, . . ., x,) which is defined in K, there exists a corresponding Q (xl, . . ., x,) as described in the previous theorem.

Both theorems are proved in ref. 7, where Q is called the projection of Q* from K* onto K. 4.2 follows immediately from 4.1 in view of the fact that on the assumptions of 4.2 every predicate Q* (xl, . . ., x,,) which is defined in K is invariant with respect to K* over K. Also, as pointed out already in ref. 7, the predicates Q (q, . . ., 2,) in 4.1 and 4.2 are themselves invariant with respect to K. For if Q (a,, . . ., a,) holds in the model M of K, then Q* (al, . . . , a,) holds in all models of K* which are extensions of R.I. Let M’ be any model of K which is an extension of M. By 3.2, M’ has an extension M* which is a model of K*. Then Q* (al,. . . a,) holds in M* and so Q (a1, . . . , a,) holds in M’. The same argument applies t o - Q, showing that Q is invariant with respect to K.

Now suppose that the conditions of theorem 4.1 are satisfied but that, in addition, K consists of axioms in prenex normal form with universal quantifiers only (but contains a t least one individual constant if n = 0). Combining 4.1 with 2.1 and taking into account that the predicate Q obtained from 4.1 is invariant with respect to K, we then see that Q may be replaced by a predicate without quantifiers whose extralogical constants all belong to K. Thus,

4.3. Theorem. Suppose that the assumptions of 4.1 hold and that K consists of axioms in prenex normal form with universal quantifiers only (but contains a t least one individual constant if n = 0). Then the conclusion of 4.1 applies with the additional condition that Q (q, . . . , xn) is free of quantifiers.

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Similarly, 4.4. As 4.3, with 4.1 replaced by 4.2 wherever i t

appears. Theorems 4.3 and 4.4 provide what is perhaps a better reali-

zation of the idea of a rational test than provided in ref. 7. The predicate Q as obtained from 4.1 and 4.2 has been called a rational test because it permits us to stay within the model M of K. How- ever, it is a fact (cf. the example of a diophantine equation) that i t may be more difficult to decide Q in M than to decide Q* in an extension of M which is a model of K*. By contrast, the predicate Q as obtained from 4.3 and 4.4 not only permits us to stay within M, but moreover, if M is given effectively, enables us to verify effectively whether or not Q holds.

Theorem.

Finally, we have 4.5. Theorem. Suppose that the set of axioms K* is an

extension of, and is model-consistent with, and model-complete relative to, a non-empty and consistent set of axioms K, where K consists of axioms in prenex normal form with universal quantifiers only, but contains a t least one individual constant if R = 0. Sup- pose further that the extralogical constants of K and K* coincide. Then to every predicate Q* (q, . . . , z,) defined in K*, there exists a predicate Q’ (q, . . ., x,,), also defined in K*, and free of quantifiers such that

Remark. Proof of 4.5. Let Q* (q, . . ., z,) be a predicate which is

defined in K* (and hence in K), and Iet Q’ (q, . . ., z,,) be a predicate which represents the projection of Q* from K* to K such that Q’ is free of quantifiers. Let M* be any model of K*, then M* is also a model of K (since I< C K*). For any a,, . . ., a,, in M*, Q* (al, . . ., a,) holds in M* if and only if Q’ (a1, . . ., a,) holds in M*. (Put M = M* in 4.4.) This proves 4.5.

Theorem 4.5 provides a satisfactory answer to the question raised in the introduction. I t proves the existence of a a method of elimination D for a wide class of theories including the theories

By the assumption of 4.5, K* is model-complete.


(i) and (ii) which were defined in section 3. As it stands, our argument does not however, provide an effective elimination procedure for the general case.

An interesting point arises when K* is model-complete but not complete, as in (i). Theorem 4.5 then affirms that to a given Q* (q, . . ., 2,) there exists aQ’ (xl, . . ., x,), free of quantifiers, such that Q’ is equivalent to Q* independently of the characteristic of the field. However, it does not follow that a predicate Q‘, which is free of quantifiers, and which is equivalent t o Q* for the algebraically closed fields of one particular characteristic, is by necessity equivalent to Q* for all algebraically closed fields. Con- sider, for instance, the predicate Q* which is given informally by

Q* (xi, ~ 2 ) = (~1) (92) “28 + 2x1~1 + x2 = 0) A ( ~ a ” + 2x1~2 + $2 = 0) 3

3 (Yl = Y2)I-

For all fields of characteristic # 2 this is equivalent to Q1 (q, x2) = ($ - x2 = 0) which is formally free of quantifiers. However, for a field of characteristic 2, Q is satisfied for all xl and x2. Thus, in order to include all characteristics, we must replace Q1 (q, x2) by

Q’ (xi, ~ 2 ) = [(G - x2 = 0) v (1 + 1 = O)]

or, formally,

Q’ (q, ~ 2 ) = [E (0 (n (xi, xi)> P ( ~ 2 ) ) 9 0) V E (0 (1, 011 Theorem 4.5 affirms that in all cases under consideration, a single predicate Q‘ can be found t o cover all models of K*.

406 A. ROBINSOIS

References

[I] HILBERT D. and BERNAYS P., Grundiagen der Mathemafik, vol. 2,

[2] KREISEL G., Absfract No. 173t, 174, 175f, Bulletin of the American

[3] JLoi J., On the edending of models I , Fundamenta Mathematicae,

[4] ROBINSON A,, Complete theories, Studies in Logic and the Foundafions

[5] ROBINSON A., On a problem of L. Henkin, Journal of Symbolic Logic,

[6] ROBINSON A., Completeness and persistence in the theory of models, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik,

171 ROBINSON A., Some problems of definability in the lower predicate calculus, Fundamenta Mathematicae, vol. 44 (1957), pp. 309-329.

[8] TARSKI A. and MCKINSEY J. C . C . , A decision method for elementary algebra and georneLry (1st edition 1948), 2nd edition, Berkeley and Los Angeles, 1951.

[9] TARSKI A., Contributions f o the theory of models, I , I I , Indagationes Mathematicae, vol. 16 (1954), pp. 572-581, 582-588.

Berlin, 1939.

Mathematical Society, vol. 63 (1957), pp. 99-100.

V O ~ . 42 (1955), pp. 38-54.

of Mathematics, Amsterdam, 1956.

V O ~ . 21 (1956), pp. 33-35.

V O ~ . 2 (1953), pp. 15-26.


Abstract

Most of the early proofs of the decidability or completeness of certain mathematical theories were based on the method of eliminations of quantifiers. Various more recent results on completeness were obtained independently of such procedures. However, it is shown in the present paper that, conversely, the completeness of a mathematical theory will in certain circumstances entail the existence of an elimination method. The proof involves the application of the extended first &-theorem of Hilbert-Bernays.

Zusammenfassung

Die meisten friiheren Beweise der Vollstandigkeit oder Entscheidbarkeit gewisser mathematischer Theorien benutzten die sogenannte Quantoren- eliminierungsmethode. Dagegen sind einige neuere Ergebnisse uber Voll- standigkeit unabhangig von diesem Verfahren. Es ist deshalb von Interesse festzustellen, dass umgekehrt die Vollstandigkeit einer mathematischen Theorie unter Umstanden die Existenz einer Eliminierungsmethode fur diese Theorie nach sich zieht. Der Beweis benutzt das erweiterte erste &-Theorem von Hilbert-Bernays.

Sommaire

Beaucoup de dCmonstrations concernant la compl6tude de certaines thCories mathkmatiques, ou concernant l’existence d’un procCd6 de dCcision pour une telle thCorie, sont bas6es sur des mCthodes d’elimination de quan- tificateurs. Or, il y a des rksultats plus rCcents sur la complCtude de certaines th6ories qui ne dCpendent pas de telles mCthodes d’klimination. Toutefois on dkmontre ici que la complCtude d’une thCorie mathbmatique entraine sous certaines conditions l’existence d‘une methode d’klimination. Au cours de la demonstration on se sert d’un des thCor&mes-s de Hilbert- Bernays.

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RELATIVE MODEL-COMPLETENESS AND THE ELIMINATION OF QUANTIFIERS